Chapter: Clustering and classification
About the Data set
Boston data set has 506 observations and 14 variables. It is one of the pre installed data sets that comes with the Mass Package. It contains information regarding the Housing values in suburbs of Boston.
The variables involved are as follows:
crim - per capita crime rate by town.
zn - proportion of residential land zoned for lots over 25,000 sq.ft.
indus - proportion of non-retail business acres per town.
chas - Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox - nitrogen oxides concentration (parts per 10 million).
rm - average number of rooms per dwelling.
age - proportion of owner-occupied units built prior to 1940.
dis - weighted mean of distances to five Boston employment centres.
rad - index of accessibility to radial highways.
tax - full-value property-tax rate per $10,000.
ptratio - pupil-teacher ratio by town.
black - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.
lstat - lower status of the population (percent).
medv - median value of owner-occupied homes in $1000s.
Source Harrison, D. and Rubinfeld, D.L. (1978) Hedonic prices and the demand for clean air. J. Environ. Economics and Management 5, 81–102. Belsley D.A., Kuh, E. and Welsch, R.E. (1980) Regression Diagnostics. Identifying Influential Data and Sources of Collinearity. New York: Wiley.
##Loading the Data Set, Installing package and Exploring Data set
#Loading required libraries for the Exercise
library(ggplot2)
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(corrplot)## corrplot 0.92 loadedlibrary(GGally)## Warning: package 'GGally' was built under R version 4.2.2## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(tidyr)
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## select#Loading Data Set
data("Boston")
#Exploring the Data Set
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Observation: The data set has 506 Observations of 14 variables The variables chas and rad are integers whiles others are numerical. From the function dimensions, from the Min. and Max range we can see that the variable distribution is skewed except for the variables rm, and medv.
##Graphical Distribution and summary of variables
# plot matrix of the variables
pairs(Boston)
# printing the correlation matrix to look at the correlations between variables in the data
cor_matrix <- cor(Boston) %>% round(digits = 2)
# visualize the correlation matrix
library(corrplot)
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6,)
Observations
Pair Plot: The pair plot is the visual representation of the observation as mentioned before this
Correlation Plot: According to the Correlation plot, we can say that there many variables with high positive correlation for example the variables ‘rad’ and ‘tax’ have the highest positive correlation among all the variables. On the other hand, we can also see strong negative correlation between many variable, highest negative correlation being among the variable like ‘lstat’ and ‘medv’, ‘age’ and ‘dis’, ‘nox’ and ‘dis’, and, ‘indus’ and ‘dis’. Therefore, we can also say that ‘dis’ (distance) has the most highest negative correlation among all the other variables
##Standardization of Data Set and Categorization of Variable
- Standardization of data Set It is the process through which we take the mean of the relevant columns, subtract them, then divide the result by the standard deviation.
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(scale(Boston))
boston_scaled## crim zn indus chas nox rm
## 1 -0.419366929 0.28454827 -1.28663623 -0.2723291 -0.14407485 0.413262920
## 2 -0.416926670 -0.48724019 -0.59279438 -0.2723291 -0.73953036 0.194082387
## 3 -0.416928995 -0.48724019 -0.59279438 -0.2723291 -0.73953036 1.281445551
## 4 -0.416338404 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.015297761
## 5 -0.412074053 -0.48724019 -1.30558569 -0.2723291 -0.83445805 1.227362043
## 6 -0.416631374 -0.48724019 -1.30558569 -0.2723291 -0.83445805 0.206891639
## 7 -0.409837246 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8 -0.403296561 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9 -0.395543302 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10 -0.400333140 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11 -0.393956378 0.04872402 -0.47618230 -0.2723291 -0.26489191 0.131459378
## 12 -0.406444832 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13 -0.409198989 0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14 -0.346886928 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15 -0.345933611 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16 -0.347162460 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17 -0.297573695 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18 -0.328932014 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19 -0.326780075 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20 -0.335721492 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21 -0.274570851 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22 -0.321045059 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23 -0.276816959 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24 -0.305188606 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25 -0.332877817 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26 -0.322382028 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27 -0.341986646 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28 -0.308985598 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29 -0.330235268 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.299402903
## 30 -0.303558666 -0.48724019 -0.43682573 -0.2723291 -0.14407485 0.554164692
## 31 -0.288635766 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32 -0.262604396 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33 -0.258736486 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34 -0.286204807 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35 -0.232598159 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36 -0.412641393 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37 -0.408773484 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38 -0.410784750 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39 -0.399750686 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40 -0.416889467 2.72854505 -1.19334657 -0.2723291 -1.09335175 0.441727925
## 41 -0.416196569 2.72854505 -1.19334657 -0.2723291 -1.09335175 1.052302267
## 42 -0.405285738 -0.48724019 -0.61611679 -0.2723291 -0.92075595 0.690796712
## 43 -0.403651148 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44 -0.401574777 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45 -0.405837965 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46 -0.400172703 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47 -0.398203290 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48 -0.393447167 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49 -0.390587216 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50 -0.394551620 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51 -0.409786092 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52 -0.415059564 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53 -0.413870242 0.41317968 -0.80123846 -0.2723291 -0.99842406 0.322174907
## 54 -0.414310861 0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55 -0.418520570 2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56 -0.418577536 3.37170210 -1.44552019 -0.2723291 -1.30909650 1.372533565
## 57 -0.417712575 3.15731641 -1.51548743 -0.2723291 -1.24868797 0.139998879
## 58 -0.418436864 3.80047346 -1.43094368 -0.2723291 -1.24005818 0.756266222
## 59 -0.402145605 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60 -0.408094536 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61 -0.402742009 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62 -0.400138988 0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63 -0.407281891 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.243896145
## 64 -0.405395021 0.58468822 -0.87557866 -0.2723291 -0.87760700 0.679410711
## 65 -0.417833484 0.26310970 -1.42219777 -0.2723291 -1.19604625 1.166162284
## 66 -0.415934988 2.94293073 -1.13212523 -0.2723291 -1.35224545 0.007636609
## 67 -0.415010735 2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68 -0.413371495 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69 -0.404344047 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70 -0.405202032 0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71 -0.409840734 -0.48724019 -0.04763292 -0.2723291 -1.22279860 0.188389387
## 72 -0.401644532 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73 -0.409447781 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74 -0.397385995 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75 -0.410921935 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76 -0.409043203 -0.48724019 0.24681257 -0.2723291 -1.01568364 0.001943608
## 77 -0.408297988 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78 -0.409979081 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79 -0.413537744 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80 -0.410351107 -0.48724019 0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81 -0.415319982 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.629596953
## 82 -0.414914241 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.475885930
## 83 -0.415847794 0.58468822 -0.91493524 -0.2723291 -1.11061133 0.024715612
## 84 -0.415973353 0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85 -0.414220179 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.148538381
## 86 -0.413434274 -0.48724019 -0.96886832 -0.2723291 -0.91212616 0.491541682
## 87 -0.414070206 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88 -0.411788058 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89 -0.413521468 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.028107013
## 90 -0.413937672 -0.48724019 -1.12629463 -0.2723291 -0.56693456 1.130581029
## 91 -0.414656148 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.188389387
## 92 -0.415530409 -0.48724019 -1.12629463 -0.2723291 -0.56693456 0.171310384
## 93 -0.415215350 0.71331963 0.56895343 -0.2723291 -0.78267931 0.223970642
## 94 -0.416759258 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95 -0.415109555 0.71331963 0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96 -0.405913532 -0.48724019 -1.20209248 -0.2723291 -0.94664532 0.484425431
## 97 -0.406727340 -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98 -0.406054205 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.539598741
## 99 -0.410583624 -0.48724019 -1.20209248 -0.2723291 -0.94664532 2.185209437
## 100 -0.412126370 -0.48724019 -1.20209248 -0.2723291 -0.94664532 1.610216351
## 101 -0.402818740 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.629596953
## 102 -0.406811046 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.706452465
## 103 -0.393506459 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.171310384
## 104 -0.395500287 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.403872039 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.404683521 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.400198280 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.404852095 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.405218308 -0.48724019 -0.37560439 -0.2723291 -0.29941107 0.269514649
## 110 -0.389452536 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.407553935 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.408378206 -0.48724019 -0.16424500 -0.2723291 -0.06640675 0.612517950
## 113 -0.405768210 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
## 114 -0.394278413 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.274166933
## 115 -0.403556979 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.043600398
## 116 -0.400182004 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.507579968
## 117 -0.404804429 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.154613915
## 118 -0.402549021 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.375217698
## 119 -0.404920687 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.587281980
## 120 -0.403272146 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.787960260
## 121 -0.412081029 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.590128481
## 122 -0.411771782 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.399412952
## 123 -0.409290833 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.460612711
## 124 -0.402618775 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.610053984
## 125 -0.408651413 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.577319229
## 126 -0.400451723 -0.48724019 2.11552109 -0.2723291 0.22700611 -0.425031456
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## 325 -0.380447177 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.185542886
## 326 -0.397796386 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.208314890
## 327 -0.384820810 -0.48724019 -0.54760720 -0.2723291 -0.53241540 0.038948114
## 328 -0.392079971 -0.48724019 -0.54760720 -0.2723291 -0.53241540 -0.286976185
## 329 -0.412408877 -0.48724019 -1.15107469 -0.2723291 -0.81719847 -0.592974981
## 330 -0.412284481 -0.48724019 -1.15107469 -0.2723291 -0.81719847 0.068836369
## 331 -0.414818909 -0.48724019 -1.15107469 -0.2723291 -0.81719847 -0.200157922
## 332 -0.414262032 1.01345959 -0.74001712 -0.2723291 -1.00791683 -0.823541516
## 333 -0.416072172 1.01345959 -0.74001712 -0.2723291 -1.00791683 -0.360985196
## 334 -0.414192278 -0.48724019 -0.86683276 -0.2723291 -0.34256002 0.044641115
## 335 -0.415755950 -0.48724019 -0.86683276 -0.2723291 -0.34256002 0.036101614
## 336 -0.415496694 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.352445695
## 337 -0.416117513 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.591551731
## 338 -0.416566270 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.554547225
## 339 -0.416258185 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.321134190
## 340 -0.413710969 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.426454706
## 341 -0.412950640 -0.48724019 -0.86683276 -0.2723291 -0.34256002 -0.450649960
## 342 -0.418589162 1.01345959 -1.40179066 -0.2723291 -0.97253469 1.361147563
## 343 -0.417197552 -0.48724019 -1.34785757 -0.2723291 -0.31667065 0.363449163
## 344 -0.417145235 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.585476196
## 345 -0.416556969 1.87100232 -1.07236154 -0.2723291 -0.61008351 0.838814735
## 346 -0.416482564 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.385180450
## 347 -0.412937852 -0.48724019 -0.98344483 -0.2723291 -0.97253469 -0.550277475
## 348 -0.417927653 3.15731641 -1.01842846 -0.2723291 -1.08472196 0.329291158
## 349 -0.418356646 2.94293073 -1.33036576 -0.2723291 -1.03294322 0.498657933
## 350 -0.416731356 1.22784527 -1.44114723 -0.2723291 -1.08472196 0.931325998
## 351 -0.412880885 1.22784527 -1.44114723 -0.2723291 -1.08472196 0.292286652
## 352 -0.410859155 2.08538800 -1.37701059 -0.2723291 -1.24005818 0.418955921
## 353 -0.411679938 2.08538800 -1.37701059 -0.2723291 -1.24005818 -0.570202978
## 354 -0.418114829 3.37170210 -1.32890811 -0.2723291 -1.24868797 0.631020203
## 355 -0.415101417 2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.884741275
## 356 -0.407709721 2.94293073 -1.34494227 -0.2723291 -1.22279860 -0.496193967
## 357 0.624240921 -0.48724019 1.01499462 3.6647712 1.85803641 -0.103376907
## 358 0.027457444 -0.48724019 1.01499462 3.6647712 1.85803641 0.157077882
## 359 0.184646645 -0.48724019 1.01499462 3.6647712 1.85803641 -0.224353176
## 360 0.075310474 -0.48724019 1.01499462 -0.2723291 1.85803641 -0.245701929
## 361 0.107933683 -0.48724019 1.01499462 -0.2723291 1.85803641 0.161347633
## 362 0.025962364 -0.48724019 1.01499462 -0.2723291 1.85803641 -0.047870149
## 363 0.007521491 -0.48724019 1.01499462 -0.2723291 1.85803641 -1.313139590
## 364 0.070785706 -0.48724019 1.01499462 3.6647712 1.85803641 -0.685486245
## 365 -0.016188203 -0.48724019 1.01499462 3.6647712 1.40928733 3.551529643
## 366 0.109555485 -0.48724019 1.01499462 -0.2723291 1.40928733 -3.876413226
## 367 0.009699007 -0.48724019 1.01499462 -0.2723291 1.40928733 -1.881016425
## 368 1.151964713 -0.48724019 1.01499462 -0.2723291 0.65849561 -3.446591661
## 369 0.149356473 -0.48724019 1.01499462 -0.2723291 0.65849561 -1.871053674
## 370 0.239079888 -0.48724019 1.01499462 3.6647712 0.65849561 0.566973944
## 371 0.340082672 -0.48724019 1.01499462 3.6647712 0.65849561 1.040916265
## 372 0.653228737 -0.48724019 1.01499462 -0.2723291 0.65849561 -0.097683907
## 373 0.541033778 -0.48724019 1.01499462 3.6647712 0.97779784 -0.583012230
## 374 0.871305835 -0.48724019 1.01499462 -0.2723291 0.97779784 -1.962141687
## 375 1.730465429 -0.48724019 1.01499462 -0.2723291 0.97779784 -3.055197852
## 376 1.859616644 -0.48724019 1.01499462 -0.2723291 1.00368721 1.463621579
## 377 1.357253412 -0.48724019 1.01499462 -0.2723291 1.00368721 0.518583436
## 378 0.721959412 -0.48724019 1.01499462 -0.2723291 1.00368721 0.724954717
## 379 2.329195069 -0.48724019 1.01499462 -0.2723291 1.00368721 0.135729129
## 380 1.657048387 -0.48724019 1.01499462 -0.2723291 1.00368721 -0.087721155
## 381 9.924109610 -0.48724019 1.01499462 -0.2723291 1.00368721 0.972600255
## 382 1.425427210 -0.48724019 1.01499462 -0.2723291 1.00368721 0.370565414
## 383 0.647964566 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.065494052
## 384 0.509089517 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.088266056
## 385 1.914932287 -0.48724019 1.01499462 -0.2723291 1.25395112 -2.727850303
## 386 1.534407630 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.434115858
## 387 2.415877170 -0.48724019 1.01499462 -0.2723291 1.25395112 -2.323647242
## 388 2.206996093 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.828356167
## 389 1.246308229 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.999146193
## 390 0.527604795 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.273288584
## 391 0.389305224 -0.48724019 1.01499462 -0.2723291 1.25395112 -0.813578764
## 392 0.195258692 -0.48724019 1.01499462 -0.2723291 1.25395112 -0.332520192
## 393 0.925923929 -0.48724019 1.01499462 -0.2723291 1.25395112 -1.777119159
## 394 0.584922404 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.130418661
## 395 1.133084385 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.565933227
## 396 0.593291831 -0.48724019 1.01499462 -0.2723291 1.19354259 0.265244898
## 397 0.262572179 -0.48724019 1.01499462 -0.2723291 1.19354259 0.171310384
## 398 0.471833420 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.765188257
## 399 4.038608880 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.183623820
## 400 0.732778398 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.615746985
## 401 2.491712382 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.423608206
## 402 1.234973056 -0.48724019 1.01499462 -0.2723291 1.19354259 0.083068871
## 403 0.695478123 -0.48724019 1.01499462 -0.2723291 1.19354259 0.169887134
## 404 2.463298882 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.331641842
## 405 4.408007629 -0.48724019 1.01499462 -0.2723291 1.19354259 -1.072610303
## 406 7.476247076 -0.48724019 1.01499462 -0.2723291 1.19354259 -0.856276271
## 407 1.988326078 -0.48724019 1.01499462 -0.2723291 0.90012973 -3.055197852
## 408 0.969311483 -0.48724019 1.01499462 -0.2723291 0.90012973 -0.963020037
## 409 0.440661113 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.950210785
## 410 1.258468834 -0.48724019 1.01499462 -0.2723291 0.36508275 0.807503230
## 411 5.524853484 -0.48724019 1.01499462 -0.2723291 0.36508275 -0.750955755
## 412 1.213407163 -0.48724019 1.01499462 -0.2723291 0.36508275 0.529969438
## 413 1.766830989 -0.48724019 1.01499462 -0.2723291 0.36508275 -2.357805247
## 414 2.911369543 -0.48724019 1.01499462 -0.2723291 0.36508275 -1.607752384
## 415 4.898256758 -0.48724019 1.01499462 -0.2723291 1.19354259 -2.512939520
## 416 1.682381045 -0.48724019 1.01499462 -0.2723291 1.07272553 0.212584640
## 417 0.839462719 -0.48724019 1.01499462 -0.2723291 1.07272553 0.707875715
## 418 2.595705326 -0.48724019 1.01499462 -0.2723291 1.07272553 -1.395688102
## 419 8.128839131 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.466305712
## 420 0.953174847 -0.48724019 1.01499462 -0.2723291 1.40928733 0.767652224
## 421 0.868899291 -0.48724019 1.01499462 -0.2723291 1.40928733 0.179849885
## 422 0.396331868 -0.48724019 1.01499462 -0.2723291 1.40928733 -0.396566452
## 423 0.980600153 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.906090028
## 424 0.399567334 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.258511181
## 425 0.602054210 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.024219796
## 426 1.423787970 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.553123975
## 427 1.003735531 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.637095738
## 428 3.958402360 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.117609410
## 429 0.436385137 -0.48724019 1.01499462 -0.2723291 1.07272553 -0.130418661
## 430 0.665620696 -0.48724019 1.01499462 -0.2723291 1.07272553 0.135729129
## 431 0.567177918 -0.48724019 1.01499462 -0.2723291 0.25289548 0.090185122
## 432 0.749723028 -0.48724019 1.01499462 -0.2723291 0.25289548 0.780461476
## 433 0.329071860 -0.48724019 1.01499462 -0.2723291 0.25289548 0.199775388
## 434 0.228743373 -0.48724019 1.01499462 -0.2723291 1.36613839 0.215431141
## 435 1.197444914 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.109069908
## 436 0.877386138 -0.48724019 1.01499462 -0.2723291 1.59914271 0.490118432
## 437 1.256434316 -0.48724019 1.01499462 -0.2723291 1.59914271 0.251012396
## 438 1.344372005 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.188771920
## 439 1.170089364 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.497617217
## 440 0.671635895 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.935978283
## 441 2.143519126 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.664137492
## 442 0.710413811 -0.48724019 1.01499462 -0.2723291 1.59914271 0.172733634
## 443 0.238660196 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.093414156
## 444 0.738590145 -0.48724019 1.01499462 -0.2723291 1.59914271 0.285170401
## 445 1.068270448 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.612900484
## 446 0.820582390 -0.48724019 1.01499462 -0.2723291 1.59914271 0.248165896
## 447 0.310937908 -0.48724019 1.01499462 -0.2723291 1.59914271 0.080222370
## 448 0.733743341 -0.48724019 1.01499462 -0.2723291 1.59914271 -0.047870149
## 449 0.664481366 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.141804663
## 450 0.454858563 -0.48724019 1.01499462 -0.2723291 1.36613839 0.188389387
## 451 0.360888236 -0.48724019 1.01499462 -0.2723291 1.36613839 0.660908458
## 452 0.212475366 -0.48724019 1.01499462 -0.2723291 1.36613839 0.527122938
## 453 0.171672232 -0.48724019 1.01499462 -0.2723291 1.36613839 0.017599361
## 454 0.538806271 -0.48724019 1.01499462 -0.2723291 1.36613839 1.577481596
## 455 0.685935651 -0.48724019 1.01499462 -0.2723291 1.36613839 0.631020203
## 456 0.132400217 -0.48724019 1.01499462 -0.2723291 1.36613839 0.342100410
## 457 0.122688009 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.439263958
## 458 0.533282845 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.496193967
## 459 0.481158489 -0.48724019 1.01499462 -0.2723291 1.36613839 0.023292362
## 460 0.370589982 -0.48724019 1.01499462 -0.2723291 1.36613839 -0.289822685
## 461 0.139347806 -0.48724019 1.01499462 -0.2723291 1.36613839 0.592592447
## 462 0.009252575 -0.48724019 1.01499462 -0.2723291 1.36613839 0.130036128
## 463 0.353587222 -0.48724019 1.01499462 -0.2723291 1.36613839 0.046064365
## 464 0.256654638 -0.48724019 1.01499462 -0.2723291 1.36613839 0.325021407
## 465 0.491283414 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.107646658
## 466 -0.052307295 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.748109254
## 467 0.018770633 -0.48724019 1.01499462 -0.2723291 0.86561057 -0.473421963
## 468 0.094024554 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.400836202
## 469 1.390700891 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.510426469
## 470 1.099985680 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.813578764
## 471 0.085480740 -0.48724019 1.01499462 -0.2723291 0.21837632 -0.167423167
## 472 0.049396526 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.079181654
## 473 -0.005213430 -0.48724019 1.01499462 -0.2723291 0.21837632 0.216854391
## 474 0.120137304 -0.48724019 1.01499462 -0.2723291 0.51178918 0.989679257
## 475 0.516449822 -0.48724019 1.01499462 -0.2723291 0.25289548 -1.220628326
## 476 0.323150830 -0.48724019 1.01499462 -0.2723291 0.25289548 -0.174539418
## 477 0.146239592 -0.48724019 1.01499462 -0.2723291 0.51178918 0.283747151
## 478 1.326491497 -0.48724019 1.01499462 -0.2723291 0.51178918 -1.395688102
## 479 0.769568300 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.141804663
## 480 1.246308229 -0.48724019 1.01499462 -0.2723291 0.51178918 -0.079181654
## 481 0.256987136 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.060679401
## 482 0.243520951 -0.48724019 1.01499462 -0.2723291 -0.19585359 0.662331708
## 483 0.246192564 -0.48724019 1.01499462 -0.2723291 -0.19585359 1.104962525
## 484 -0.092441945 -0.48724019 1.01499462 -0.2723291 -0.19585359 -0.743839504
## 485 -0.143573456 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.588705230
## 486 0.006992516 -0.48724019 1.01499462 -0.2723291 0.24426569 0.038948114
## 487 0.241610829 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.242855428
## 488 0.142084524 -0.48724019 1.01499462 -0.2723291 0.24426569 -0.540314723
## 489 -0.402562972 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.182200570
## 490 -0.398783418 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.239130578
## 491 -0.395982758 -0.48724019 2.42017014 -0.2723291 0.46864023 -1.695993897
## 492 -0.407808541 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 493 -0.407159820 -0.48724019 2.42017014 -0.2723291 0.46864023 -0.429301206
## 494 -0.399952975 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.822118266
## 495 -0.387599381 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.510426469
## 496 -0.399292629 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.874778524
## 497 -0.386433311 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.273288584
## 498 -0.388900310 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.698295497
## 499 -0.392302024 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.378064199
## 500 -0.399427488 -0.48724019 -0.21088983 -0.2723291 0.26152527 -1.018526795
## 501 -0.394015670 -0.48724019 -0.21088983 -0.2723291 0.26152527 -0.366678197
## 502 -0.412820431 -0.48724019 0.11562398 -0.2723291 0.15796779 0.438881424
## 503 -0.414838673 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.234315927
## 504 -0.413037834 -0.48724019 0.11562398 -0.2723291 0.15796779 0.983986256
## 505 -0.407360947 -0.48724019 0.11562398 -0.2723291 0.15796779 0.724954717
## 506 -0.414589880 -0.48724019 0.11562398 -0.2723291 0.15796779 -0.362408446
## age dis rad tax ptratio black
## 1 -0.119894767 0.1400749840 -0.9818712 -0.66594918 -1.45755797 0.440615895
## 2 0.366803426 0.5566090496 -0.8670245 -0.98635338 -0.30279450 0.440615895
## 3 -0.265548971 0.5566090496 -0.8670245 -0.98635338 -0.30279450 0.396035074
## 4 -0.809087830 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.415751408
## 5 -0.510674339 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.440615895
## 6 -0.350809969 1.0766711351 -0.7521778 -1.10502160 0.11292035 0.410165113
## 7 -0.070159185 0.8384142195 -0.5224844 -0.57694801 -1.50374851 0.426376321
## 8 0.977840575 1.0236248974 -0.5224844 -0.57694801 -1.50374851 0.440615895
## 9 1.116389695 1.0861216287 -0.5224844 -0.57694801 -1.50374851 0.328123258
## 10 0.615481336 1.3283202075 -0.5224844 -0.57694801 -1.50374851 0.328999540
## 11 0.913894827 1.2117799501 -0.5224844 -0.57694801 -1.50374851 0.392639483
## 12 0.508905089 1.1547920492 -0.5224844 -0.57694801 -1.50374851 0.440615895
## 13 -1.050660656 0.7863652700 -0.5224844 -0.57694801 -1.50374851 0.370513376
## 14 -0.240681180 0.4333252240 -0.6373311 -0.60068166 1.17530274 0.440615895
## 15 0.565745754 0.3166899868 -0.6373311 -0.60068166 1.17530274 0.255720500
## 16 -0.428965883 0.3341187865 -0.6373311 -0.60068166 1.17530274 0.426595391
## 17 -1.395257187 0.3341187865 -0.6373311 -0.60068166 1.17530274 0.330533033
## 18 0.466274590 0.2198105553 -0.6373311 -0.60068166 1.17530274 0.329437681
## 19 -1.135921653 0.0006920764 -0.6373311 -0.60068166 1.17530274 -0.741378303
## 20 0.032864520 0.0006920764 -0.6373311 -0.60068166 1.17530274 0.375442459
## 21 1.048891406 0.0013569352 -0.6373311 -0.60068166 1.17530274 0.217930861
## 22 0.732715207 0.1031753182 -0.6373311 -0.60068166 1.17530274 0.392749018
## 23 0.821528746 0.0863638874 -0.6373311 -0.60068166 1.17530274 0.440615895
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## 414 1.116389695 -1.0474578907 1.6596029 1.52941294 0.80577843 -1.595971828
## 415 1.116389695 -1.0147848276 1.6596029 1.52941294 0.80577843 -2.939968567
## 416 1.116389695 -0.9309651233 1.6596029 1.52941294 0.80577843 -3.608352275
## 417 0.789555872 -0.9381835908 1.6596029 1.52941294 0.80577843 -3.670568261
## 418 0.729162665 -1.0198662487 1.6596029 1.52941294 0.80577843 -2.511795523
## 419 1.116389695 -0.9462093868 1.6596029 1.52941294 0.80577843 -3.726650277
## 420 0.281542429 -0.9502935197 1.6596029 1.52941294 0.80577843 -3.376137680
## 421 1.116389695 -0.9194725633 1.6596029 1.52941294 0.80577843 -0.415401588
## 422 0.949420242 -0.9120166463 1.6596029 1.52941294 0.80577843 -0.401928760
## 423 0.675874542 -0.8756393696 1.6596029 1.52941294 0.80577843 -0.713337295
## 424 0.587061003 -0.8421114879 1.6596029 1.52941294 0.80577843 -3.879232792
## 425 0.071942477 -0.8223081923 1.6596029 1.52941294 0.80577843 -3.866855315
## 426 0.952972784 -0.8953951752 1.6596029 1.52941294 0.80577843 -3.822712635
## 427 -0.315284553 -0.8536040479 1.6596029 1.52941294 0.80577843 -3.636831424
## 428 0.359698343 -0.9175729666 1.6596029 1.52941294 0.80577843 -3.700690438
## 429 0.338383094 -0.8830477967 1.6596029 1.52941294 0.80577843 -2.847301799
## 430 0.960077867 -0.8675660836 1.6596029 1.52941294 0.80577843 -3.241738006
## 431 0.622586419 -0.8274371034 1.6596029 1.52941294 0.80577843 -2.992764527
## 432 0.913894827 -0.8105781827 1.6596029 1.52941294 0.80577843 -3.015985986
## 433 0.221149222 -0.7572944954 1.6596029 1.52941294 0.80577843 -2.833938506
## 434 0.686532167 -0.7024911307 1.6596029 1.52941294 0.80577843 -2.809402625
## 435 0.938762618 -0.7469416934 1.6596029 1.52941294 0.80577843 -2.804583076
## 436 0.924552451 -0.7932443629 1.6596029 1.52941294 0.80577843 -2.703591634
## 437 0.878369411 -0.8512295520 1.6596029 1.52941294 0.80577843 -3.605723431
## 438 1.116389695 -0.8932106390 1.6596029 1.52941294 0.80577843 -3.804748865
## 439 0.686532167 -0.9376612017 1.6596029 1.52941294 0.80577843 -3.151590547
## 440 0.899684660 -0.9392758589 1.6596029 1.52941294 0.80577843 0.440615895
## 441 0.846396537 -0.9160057994 1.6596029 1.52941294 0.80577843 0.380919218
## 442 1.016918532 -0.8215483536 1.6596029 1.52941294 0.80577843 0.320784401
## 443 1.116389695 -0.8501847738 1.6596029 1.52941294 0.80577843 0.427362137
## 444 1.116389695 -0.8627221120 1.6596029 1.52941294 0.80577843 0.329218610
## 445 0.995603282 -0.9020437636 1.6596029 1.52941294 0.80577843 -1.272295352
## 446 0.931657534 -0.8582105699 1.6596029 1.52941294 0.80577843 -3.435177145
## 447 0.988498199 -0.8182715493 1.6596029 1.52941294 0.80577843 -0.423507192
## 448 0.995603282 -0.7584342534 1.6596029 1.52941294 0.80577843 0.348825409
## 449 1.070206655 -0.7282306659 1.6596029 1.52941294 0.80577843 0.440615895
## 450 1.055996489 -0.7646079427 1.6596029 1.52941294 0.80577843 -0.574665749
## 451 0.853501620 -0.6987869171 1.6596029 1.52941294 0.80577843 -3.903330533
## 452 1.052443947 -0.6837801032 1.6596029 1.52941294 0.80577843 -0.015160016
## 453 0.825081288 -0.6776064140 1.6596029 1.52941294 0.80577843 0.311254840
## 454 1.091521905 -0.6374774338 1.6596029 1.52941294 0.80577843 0.210263398
## 455 0.906789743 -0.6168668096 1.6596029 1.52941294 0.80577843 -3.833666154
## 456 0.636796585 -0.6455032298 1.6596029 1.52941294 0.80577843 -3.349082489
## 457 0.686532167 -0.5767378294 1.6596029 1.52941294 0.80577843 -3.792042783
## 458 0.416539008 -0.4824228534 1.6596029 1.52941294 0.80577843 -3.868498343
## 459 0.537325421 -0.4805707466 1.6596029 1.52941294 0.80577843 -0.925178346
## 460 0.562193212 -0.5117241325 1.6596029 1.52941294 0.80577843 0.440615895
## 461 0.761135540 -0.5687120333 1.6596029 1.52941294 0.80577843 -1.111169093
## 462 0.704294875 -0.5831489682 1.6596029 1.52941294 0.80577843 0.380700148
## 463 0.512457630 -0.5036983364 1.6596029 1.52941294 0.80577843 0.440615895
## 464 0.757582998 -0.4717851119 1.6596029 1.52941294 0.80577843 0.406879058
## 465 -0.112789684 -0.3949464255 1.6596029 1.52941294 0.80577843 0.440615895
## 466 -0.723826832 -0.3459843207 1.6596029 1.52941294 0.80577843 -0.243979021
## 467 0.572850837 -0.4385896596 1.6596029 1.52941294 0.80577843 -3.665748713
## 468 0.920999910 -0.5958762661 1.6596029 1.52941294 0.80577843 -0.278044464
## 469 0.086152643 -0.4210658801 1.6596029 1.52941294 0.80577843 0.132164810
## 470 -0.421860800 -0.4612898402 1.6596029 1.52941294 0.80577843 0.440615895
## 471 0.547983046 -0.3617034834 1.6596029 1.52941294 0.80577843 0.440615895
## 472 0.786003330 -0.3304076278 1.6596029 1.52941294 0.80577843 0.423418871
## 473 0.228254306 -0.4267171803 1.6596029 1.52941294 0.80577843 0.401949974
## 474 -0.034633770 -0.5993905200 1.6596029 1.52941294 0.80577843 0.197228711
## 475 0.952972784 -0.6483526248 1.6596029 1.52941294 0.80577843 -0.044844052
## 476 1.024023615 -0.7546350600 1.6596029 1.52941294 0.80577843 -0.590548351
## 477 0.889027036 -0.7074775720 1.6596029 1.52941294 0.80577843 0.433057967
## 478 1.020471073 -0.8046419430 1.6596029 1.52941294 0.80577843 -0.078799960
## 479 0.999155824 -0.7714939807 1.6596029 1.52941294 0.80577843 0.252215374
## 480 0.690084708 -0.8756393696 1.6596029 1.52941294 0.80577843 0.291867112
## 481 -0.137657475 -0.1761128861 1.6596029 1.52941294 0.80577843 0.440615895
## 482 0.224701764 -0.2200410597 1.6596029 1.52941294 0.80577843 0.398663919
## 483 0.299305137 -0.1825715149 1.6596029 1.52941294 0.80577843 0.422871195
## 484 -1.004477616 0.1440166471 1.6596029 1.52941294 0.80577843 0.397020891
## 485 -0.947636951 -0.0337381137 1.6596029 1.52941294 0.80577843 0.153962312
## 486 -0.592382795 0.0933923952 1.6596029 1.52941294 0.80577843 0.349920761
## 487 0.398776300 -0.1183176566 1.6596029 1.52941294 0.80577843 0.394392046
## 488 -0.546199754 -0.3052379716 1.6596029 1.52941294 0.80577843 0.345539353
## 489 0.857054162 -0.9375187319 -0.6373311 1.79641644 0.75958789 0.420790026
## 490 1.055996489 -0.9686246278 -0.6373311 1.79641644 0.75958789 -0.138277566
## 491 1.045338864 -0.9367114033 -0.6373311 1.79641644 0.75958789 -0.418906714
## 492 1.073759197 -0.9151034909 -0.6373311 1.79641644 0.75958789 0.366241503
## 493 0.530220338 -0.8002728706 -0.6373311 1.79641644 0.75958789 0.440615895
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512 0.34387304 0.440615895
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512 0.34387304 0.440615895
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512 0.34387304 0.401073693
## 497 0.153650933 -0.4732098094 -0.4076377 -0.10227512 0.34387304 0.440615895
## 498 0.071942477 -0.4285217972 -0.4076377 -0.10227512 0.34387304 0.440615895
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512 0.34387304 0.440615895
## 500 0.174966182 -0.6625521101 -0.4076377 -0.10227512 0.34387304 0.428238419
## 501 0.395223759 -0.6158695213 -0.4076377 -0.10227512 0.34387304 0.440615895
## 502 0.018654354 -0.6251775451 -0.9818712 -0.80241764 1.17530274 0.386834118
## 503 0.288647512 -0.7159307773 -0.9818712 -0.80241764 1.17530274 0.440615895
## 504 0.796660955 -0.7729186782 -0.9818712 -0.80241764 1.17530274 0.440615895
## 505 0.736267749 -0.6677760011 -0.9818712 -0.80241764 1.17530274 0.402826256
## 506 0.434301716 -0.6126402069 -0.9818712 -0.80241764 1.17530274 0.440615895
## lstat medv
## 1 -1.074498970 0.159527789
## 2 -0.491952525 -0.101423917
## 3 -1.207532413 1.322937477
## 4 -1.360170785 1.181588636
## 5 -1.025486649 1.486032293
## 6 -1.042290874 0.670558212
## 7 -0.031236706 0.039924923
## 8 0.909799859 0.496590409
## 9 2.419379350 -0.655946292
## 10 0.622727693 -0.394994586
## 11 1.091845624 -0.819041108
## 12 0.086392864 -0.394994586
## 13 0.428078760 -0.090550930
## 14 -0.615183504 -0.231899770
## 15 -0.335113097 -0.471105500
## 16 -0.585776111 -0.286264709
## 17 -0.850442645 0.061670899
## 18 0.282442149 -0.547216415
## 19 -0.134862757 -0.253645746
## 20 -0.192277190 -0.471105500
## 21 1.171665689 -0.971262937
## 22 0.164812578 -0.318883672
## 23 0.849584722 -0.797295133
## 24 1.012025558 -0.873406047
## 25 0.510699530 -0.753803182
## 26 0.540106923 -0.938643973
## 27 0.302047077 -0.645073304
## 28 0.647934029 -0.840787084
## 29 0.020576319 -0.449359525
## 30 -0.094252548 -0.166661844
## 31 1.392921311 -1.069119826
## 32 0.054184768 -0.873406047
## 33 2.108501199 -1.014754888
## 34 0.797771697 -1.025627875
## 35 1.076441751 -0.982135924
## 36 -0.416333515 -0.394994586
## 37 -0.174072614 -0.275391721
## 38 -0.543765550 -0.166661844
## 39 -0.353317674 0.235638703
## 40 -1.166922204 0.898890955
## 41 -1.494604580 1.344683452
## 42 -1.094103899 0.442225470
## 43 -0.958269752 0.300876629
## 44 -0.730012370 0.235638703
## 45 -0.434538092 -0.144915868
## 46 -0.342114857 -0.351502635
## 47 0.209623843 -0.275391721
## 48 0.860787538 -0.645073304
## 49 2.542610329 -0.884279035
## 50 0.496696010 -0.340629648
## 51 0.111599201 -0.308010684
## 52 -0.451342316 -0.221026782
## 53 -1.032488409 0.268257666
## 54 -0.591377519 0.094289862
## 55 0.300646725 -0.394994586
## 56 -1.098304955 1.399048391
## 57 -0.963871160 0.235638703
## 58 -1.218735230 0.985874857
## 59 -0.811232788 0.083416874
## 60 -0.480749709 -0.318883672
## 61 0.069588640 -0.416740562
## 62 0.250234052 -0.710311231
## 63 -0.829437365 -0.036185991
## 64 -0.441539852 0.268257666
## 65 -0.644590896 1.138096685
## 66 -1.117909883 0.105162850
## 67 -0.337913801 -0.340629648
## 68 -0.637589136 -0.057931966
## 69 0.061186528 -0.558089402
## 70 -0.540964846 -0.177534831
## 71 -0.830837717 0.181273764
## 72 -0.388326475 -0.090550930
## 73 -0.998879961 0.029051936
## 74 -0.716008850 0.094289862
## 75 -0.822435605 0.170400776
## 76 -0.519959566 -0.123169893
## 77 -0.095652900 -0.275391721
## 78 -0.333712745 -0.188407819
## 79 -0.043839875 -0.144915868
## 80 -0.497553933 -0.242772758
## 81 -1.031088057 0.594447298
## 82 -0.760820115 0.148654801
## 83 -0.830837717 0.246511690
## 84 -0.720209906 0.039924923
## 85 -0.424735627 0.148654801
## 86 -0.857444405 0.442225470
## 87 0.028978431 -0.003567028
## 88 -0.589977167 -0.036185991
## 89 -1.001680665 0.116035838
## 90 -0.973673624 0.670558212
## 91 -0.538164142 0.007305960
## 92 -0.623585616 -0.057931966
## 93 -0.629187024 0.039924923
## 94 -0.902255670 0.268257666
## 95 -0.288901480 -0.210153795
## 96 -0.840640181 0.637939249
## 97 -0.183875078 -0.123169893
## 98 -1.182326077 1.757856987
## 99 -1.271948607 2.312379361
## 100 -0.905056374 1.159842661
## 101 -0.452742668 0.540082359
## 102 -0.697804274 0.431352482
## 103 -0.283300072 -0.427613550
## 104 0.110198849 -0.351502635
## 105 -0.045240227 -0.264518733
## 106 0.534505515 -0.329756660
## 107 0.841182610 -0.329756660
## 108 0.201221731 -0.231899770
## 109 -0.053642339 -0.297137697
## 110 0.405673128 -0.340629648
## 111 0.048583360 -0.090550930
## 112 -0.349116618 0.029051936
## 113 0.498096362 -0.405867574
## 114 0.621327341 -0.416740562
## 115 -0.308506409 -0.438486537
## 116 0.435080520 -0.460232513
## 117 -0.085850436 -0.144915868
## 118 -0.329511689 -0.362375623
## 119 0.380466791 -0.231899770
## 120 0.134004834 -0.351502635
## 121 0.240431588 -0.057931966
## 122 0.226428068 -0.242772758
## 123 0.738956911 -0.221026782
## 124 1.786420232 -0.568962390
## 125 0.689944590 -0.405867574
## 126 0.302047077 -0.123169893
## 127 2.045485358 -0.742930194
## 128 0.635330861 -0.688565255
## 129 0.383267495 -0.492851476
## 130 0.796371345 -0.895152022
## 131 -0.007430722 -0.362375623
## 132 -0.055042691 -0.318883672
## 133 -0.214682823 0.050797911
## 134 0.332854822 -0.449359525
## 135 0.652135085 -0.753803182
## 136 0.603122764 -0.481978488
## 137 0.594720652 -0.558089402
## 138 0.271239333 -0.590708366
## 139 1.213676250 -1.003881900
## 140 0.813175569 -0.514597451
## 141 1.611376228 -0.927770986
## 142 3.046737061 -0.884279035
## 143 1.983869868 -0.993008912
## 144 1.927855787 -0.753803182
## 145 2.329756820 -1.166976716
## 146 2.121104367 -0.949516961
## 147 0.559711851 -0.753803182
## 148 2.363365269 -0.862533059
## 149 2.193922673 -0.514597451
## 150 1.231880827 -0.775549157
## 151 0.202622083 -0.112296905
## 152 0.087793216 -0.318883672
## 153 -0.074647619 -0.786422145
## 154 0.439281577 -0.340629648
## 155 0.345457990 -0.601581353
## 156 0.331454470 -0.753803182
## 157 0.488293898 -1.025627875
## 158 -1.129112700 2.040554668
## 159 -0.871447926 0.192146752
## 160 -0.737014131 0.083416874
## 161 -1.001680665 0.485717421
## 162 -1.529613381 2.986504601
## 163 -1.503006692 2.986504601
## 164 -1.306957408 2.986504601
## 165 -0.141864517 0.018178948
## 166 -0.398128939 0.268257666
## 167 -1.253744030 2.986504601
## 168 -0.071846915 0.137781813
## 169 -0.217483527 0.137781813
## 170 -0.186675782 -0.025313003
## 171 0.248833700 -0.558089402
## 172 -0.087250788 -0.373248611
## 173 0.285242853 0.061670899
## 174 -0.505956045 0.116035838
## 175 -0.421934923 0.007305960
## 176 -1.025486649 0.746669127
## 177 -0.356118378 0.072543887
## 178 -0.891052854 0.224765715
## 179 -0.802830676 0.801034065
## 180 -1.066096858 1.594762170
## 181 -0.713208146 1.877459852
## 182 -0.448541612 1.486032293
## 183 -1.096904603 1.670873085
## 184 -0.976474328 1.083731747
## 185 0.185817859 0.420479494
## 186 0.069588640 0.768415102
## 187 -1.148717628 2.986504601
## 188 -0.836439125 1.029366808
## 189 -1.133313756 0.790161078
## 190 -1.017084537 1.344683452
## 191 -1.057694746 1.573016195
## 192 -1.115109179 0.866271992
## 193 -1.369973249 1.507778268
## 194 -1.067497210 0.931509918
## 195 -1.158520092 0.714050163
## 196 -1.355969729 2.986504601
## 197 -1.200530653 1.170715648
## 198 -0.566171183 0.844526016
## 199 -0.844841237 1.312064489
## 200 -1.133313756 1.344683452
## 201 -1.148717628 1.127223698
## 202 -0.731412722 0.170400776
## 203 -1.336364800 2.149284545
## 204 -1.238340158 2.823409785
## 205 -1.368572897 2.986504601
## 206 -0.249691623 0.007305960
## 207 -0.235688103 0.203019739
## 208 0.757161488 -0.003567028
## 209 0.281041797 0.203019739
## 210 1.461538560 -0.275391721
## 211 0.646533677 -0.090550930
## 212 1.586169891 -0.351502635
## 213 0.472890025 -0.014440015
## 214 -0.458344076 0.605320286
## 215 2.366165973 0.126908825
## 216 -0.445740908 0.268257666
## 217 0.120001313 0.083416874
## 218 -0.414933163 0.670558212
## 219 0.737556559 -0.112296905
## 220 -0.301504649 0.050797911
## 221 -0.412132459 0.453098458
## 222 1.233281179 -0.090550930
## 223 -0.381324714 0.540082359
## 224 -0.707606738 0.822780041
## 225 -1.192128541 2.421109239
## 226 -1.123511291 2.986504601
## 227 -1.333564096 1.638254121
## 228 -0.881250390 0.985874857
## 229 -1.222936286 2.627696006
## 230 -1.245341918 0.975001869
## 231 -0.140464165 0.192146752
## 232 -1.036689465 0.996747845
## 233 -1.425987330 2.084046619
## 234 -1.218735230 2.801663810
## 235 -0.644590896 0.703177176
## 236 -0.248291271 0.159527789
## 237 -0.435938444 0.279130654
## 238 -1.109507771 0.975001869
## 239 -0.881250390 0.126908825
## 240 -0.739814835 0.083416874
## 241 -0.178273670 -0.057931966
## 242 -0.035437762 -0.264518733
## 243 -0.200679302 -0.036185991
## 244 -1.045091578 0.126908825
## 245 -0.021434242 -0.536343427
## 246 0.813175569 -0.438486537
## 247 -0.489151821 0.192146752
## 248 -0.350516970 -0.221026782
## 249 -0.438739148 0.213892727
## 250 -0.853243349 0.398733519
## 251 -0.945666583 0.203019739
## 252 -1.269147903 0.246511690
## 253 -1.277550015 0.768415102
## 254 -1.276149663 2.203649484
## 255 -0.851842997 -0.068804954
## 256 -0.476548653 -0.177534831
## 257 -1.336364800 2.334125337
## 258 -1.054894042 2.986504601
## 259 -0.681000049 1.464286318
## 260 -0.805631380 0.822780041
## 261 -0.428936683 1.225080587
## 262 -0.755218707 2.236268447
## 263 -0.944266231 2.856028748
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## 375 3.545262384 -0.949516961
## 376 0.110198849 -0.819041108
## 377 1.482543841 -0.938643973
## 378 1.202473434 -1.003881900
## 379 1.545559682 -1.025627875
## 380 1.278092444 -1.340944520
## 381 0.638131565 -1.319198544
## 382 1.180067802 -1.264833606
## 383 1.532956514 -1.221341655
## 384 1.667390309 -1.112611777
## 385 2.517403992 -1.493166348
## 386 2.542610329 -1.667134152
## 387 2.188321265 -1.308325557
## 388 2.707851869 -1.645388177
## 389 2.516003640 -1.340944520
## 390 1.147859705 -1.199595679
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## 392 0.855186130 0.072543887
## 393 1.824229736 -1.395309459
## 394 0.352459751 -0.949516961
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## 404 0.996621685 -1.547531287
## 405 2.062289582 -1.525785311
## 406 1.446134688 -1.906339882
## 407 1.496547361 -1.156103728
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## 410 0.998022037 0.540082359
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## 412 1.199672730 -0.579835378
## 413 3.041135653 -0.503724464
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## 415 3.406627533 -1.688880128
## 416 2.296148371 -1.667134152
## 417 1.839633609 -1.634515189
## 418 1.958663532 -1.319198544
## 419 1.115651608 -1.493166348
## 420 1.412526239 -1.536658299
## 421 0.331454470 -0.634200317
## 422 0.426678408 -0.906025010
## 423 0.202622083 -0.188407819
## 424 1.489545601 -0.993008912
## 425 0.631129805 -1.177849704
## 426 1.643584324 -1.547531287
## 427 0.425278056 -1.340944520
## 428 0.261436868 -1.264833606
## 429 1.241683291 -1.253960618
## 430 1.600173411 -1.417055434
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## 432 0.985418869 -0.916897998
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## 435 0.352459751 -1.177849704
## 436 1.486744897 -0.993008912
## 437 0.755761136 -1.406182446
## 438 1.932056843 -1.504039336
## 439 2.992123331 -1.536658299
## 440 1.432131167 -1.058246839
## 441 1.324304061 -1.308325557
## 442 0.961612885 -0.590708366
## 443 0.551309739 -0.449359525
## 444 0.867789298 -0.775549157
## 445 1.559563202 -1.275706593
## 446 1.586169891 -1.166976716
## 447 0.719351983 -0.829914096
## 448 0.530304459 -1.079992814
## 449 0.766963952 -0.916897998
## 450 0.932205492 -1.036500863
## 451 0.670339662 -0.993008912
## 452 0.710949871 -0.797295133
## 453 0.646533677 -0.699438243
## 454 0.572315020 -0.514597451
## 455 0.848184370 -0.829914096
## 456 0.766963952 -0.916897998
## 457 0.890194931 -1.069119826
## 458 0.600322060 -0.982135924
## 459 0.500897066 -0.829914096
## 460 0.286643205 -0.275391721
## 461 0.527503755 -0.666819280
## 462 0.279641445 -0.525470439
## 463 0.187218211 -0.329756660
## 464 -0.330912041 -0.253645746
## 465 0.079391104 -0.123169893
## 466 0.206823139 -0.286264709
## 467 0.629729453 -0.384121599
## 468 1.213676250 -0.373248611
## 469 0.766963952 -0.373248611
## 470 0.295045317 -0.264518733
## 471 0.509299178 -0.286264709
## 472 0.030378783 -0.318883672
## 473 0.239031236 0.072543887
## 474 -0.139063813 0.790161078
## 475 0.768364304 -0.949516961
## 476 1.602974115 -1.003881900
## 477 0.843983314 -0.634200317
## 478 1.716402630 -1.145230740
## 479 0.752960432 -0.862533059
## 480 0.063987232 -0.123169893
## 481 -0.267896200 0.050797911
## 482 -0.688001809 0.126908825
## 483 -0.790227508 0.268257666
## 484 -0.312707465 -0.079677942
## 485 0.096195329 -0.210153795
## 486 -0.290301832 -0.144915868
## 487 0.325853062 -0.373248611
## 488 -0.168471205 -0.210153795
## 489 0.757161488 -0.797295133
## 490 1.584769539 -1.688880128
## 491 2.384370549 -1.569277262
## 492 0.758561840 -0.971262937
## 493 0.097595681 -0.264518733
## 494 -0.090051492 -0.079677942
## 495 0.131204129 0.213892727
## 496 0.692745294 0.061670899
## 497 1.188469914 -0.308010684
## 498 0.202622083 -0.460232513
## 499 0.037380543 -0.144915868
## 500 0.342657286 -0.547216415
## 501 0.234830180 -0.623327329
## 502 -0.417733867 -0.014440015
## 503 -0.500354637 -0.210153795
## 504 -0.982075736 0.148654801
## 505 -0.864446166 -0.057931966
## 506 -0.668396881 -1.156103728Observation We can see that now all variables are normally distributed and because of the standardization process, it is done in such a way that mean of all variables is equal to zero
2.Creating a categorical variable of the crime rate in the Boston data set
# Create a quantile vector of crim and create the categorical "crime".
bins <- quantile(boston_scaled$crim)
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c('low','med_low','med_high','high'))
# Remove the original unscaled variable and add the new categorical value to scaled data.
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)
# Look at the table of the new factor crime
table(boston_scaled$crim)##
## low med_low med_high high
## 127 126 126 127Observation We have now created a categorical variable(previously continuous) and changed the name from crim to crime which has now been added as a separate variable in the boston_scaled data set which has 127 observation for low crime, 126 for medium to low and medium to high crime and 127 for high crime
##Divide the dataset to train and test sets.
When we want to use a statistical method to predict something, it is important to have data to test how well the predictions fit. Splitting the original data to test and train sets allows us to check how well our model works. The training of the model is done with the train set and prediction on new data is done with the test set. This way you have true classes / labels for the test data, and you can calculate how well the model performed in prediction.
Below the size = n x 0.8 representation means that it will randomly choose the 80% of the data to create a training data set
# Number of rows in the Boston dataset
n <- nrow(boston_scaled)
# Choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# Create train set
train <- boston_scaled[ind,]
# Create test set
test <- boston_scaled[-ind,]
# Save the correct classes from the test data
correct_classes <- test$crime
# Remove the crime variable from the test data
test <- dplyr::select(test, -crime)##Linear Discriminant Analysis on the train set
LDA is done to find the linear combination among the variable which separates the classes of target variables For this analysis, we will use ‘Crime’ as the target variable
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2623762 0.2425743 0.2326733 0.2623762
##
## Group means:
## zn indus chas nox rm age
## low 0.94954839 -0.9329221 -0.16090170 -0.8754251 0.43431628 -0.8687437
## med_low -0.08647022 -0.3402043 -0.15180559 -0.5790955 -0.14228392 -0.3969205
## med_high -0.38597716 0.1632302 0.18839543 0.3581973 0.09749821 0.4606812
## high -0.48724019 1.0170298 -0.01233188 1.0650727 -0.41117491 0.8251818
## dis rad tax ptratio black lstat
## low 0.8670784 -0.6806695 -0.7298733 -0.4531316 0.37912635 -0.77545776
## med_low 0.3789958 -0.5470944 -0.5400761 -0.1415989 0.31592126 -0.14497959
## med_high -0.3908764 -0.4015288 -0.2916393 -0.2059909 0.08397632 0.05043063
## high -0.8489666 1.6390172 1.5146914 0.7818116 -0.79908681 0.92898204
## medv
## low 0.51536170
## med_low 0.02117457
## med_high 0.13072594
## high -0.74785381
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.09336769 0.654029367 -0.95845199
## indus 0.05974558 -0.316941158 0.52079884
## chas -0.09923343 -0.103575292 -0.04742673
## nox 0.39770147 -0.601124811 -1.22602246
## rm -0.11301683 -0.110740555 -0.21496145
## age 0.23034316 -0.351487376 -0.28174231
## dis -0.06546022 -0.217651396 0.25404685
## rad 3.11340256 1.157357324 0.44683581
## tax 0.08946623 -0.180815893 -0.05231663
## ptratio 0.07200659 -0.005939453 -0.40499343
## black -0.10702093 0.018757572 0.13303557
## lstat 0.24462559 -0.299230554 0.35477314
## medv 0.17151119 -0.384649325 -0.16613969
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9526 0.0358 0.0116# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(factor(train$crime))
# plot the lda results
plot(lda.fit, col = classes, pch = classes, dimen = 2)
lda.arrows(lda.fit, myscale = 1)
Observation The proportion of trace tells us that LD1 explains variance up-to 96.27%, LD2 does it up to 2.84% and LD3 does it for below 1% i.e., at around 0.89%. Through the plot we can say that the target variable crime is clearly separated, and the variable accessibility to ‘rad’ is optimal as seen through the arrow.
##Prediction and Cross tabulation
(Please Note:The saving of crime categories and removal of the categorical crime variable from the test data set can be found just before the start of LDA exercise in the Divide the data set to train and test sets exercise)
# Predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# Cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 13 8 0 0
## med_low 5 12 11 0
## med_high 0 10 21 1
## high 0 0 0 21Observation The cross tabulation suggests that the model predicts the ‘high’ crime rate quite well. There are 102 total observation and The majority of the predicted values are predicted as correct values for the same category making the model usable for rough predictions.
##Reload and standardize the Boston dataset
# Reload Boston dataset.
data("Boston")
# center and standardize variables
boston_scaled <- scale(Boston)
# change the object to data frame from matrix type.
boston_scaled <- as.data.frame(boston_scaled)##Calculate the distance between Observations, K means clustering and visualisation
# Calculate the Euclidean distances between observations.
dist_eu <- dist(boston_scaled, method = "euclidean")
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970Observation The most normal distance measure is Euclidean, according to which the distance range is between 0.1343 to 14.3970, with a mean of 4.9111 and median of 4.8241
# K-means clustering
km <-kmeans(boston_scaled, centers = "3")
# plot the Boston dataset with clusters
pairs(boston_scaled[6:10], col = km$cluster)
Observation As I have clustered the data into 5 columns and definde k-means as 3 cluster
We will now find the optimal number for clusters
# Investigate optimal number of clusters and rerun algorithm
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# Visualize the cluster.
qplot(x = 1:k_max, y = twcss, geom = 'line')
Observation We can see that optimal number drops radically and with this we can see that after every two cluster, there seems to be a change.
Let us now check how does it look with optimal number 2
# k-means clustering
km <-kmeans(dist_eu, centers = 2)
# Plot the clusters
pairs(boston_scaled[6:10], col = km$cluster)
Observation We can see that because of our previous observation was of 3 clusters and optimal is 2, there isnt much difference but going by the results, ‘two’ seems to be the optimal number for clustering the data, when the method chosen is Euclidean
##Super Bonus: Create a 3-D plot
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)## Warning: package 'plotly' was built under R version 4.2.2##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layout# Plot with crime classes as color of the training data.
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)# k-means with 4 clusters to compare the methods.
km3D <-kmeans(boston_scaled, centers = 4)
#Plot with km cluster as color of data
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km3D$cluster[ind])